Midterm (Due October 27.)
Math. 610:600
(Last modified on 2015-10-16 11:15)
The midterm involves developing code to implement a two dimensional finite
element approximation. You are to do this assignment individually
and are not allowed to discuss this with other members of the class
or Wenyu.
The procedure involves changing variables mapping the local stiffness integrals on the triangles τ to integrals on the reference element τ̂ . These
are then subsequently approximated using a fixed quadrature on the reference element. It is only necessary to compute shape functions (local basis
functions) on the reference element.
Let τ̂ denote the reference triangle with vertexes (0,0), (1,0) and (0,1). We
shall use the quadrature
Z
1
g(x) da ≈
g(0, 0) + g(1, 0) + g(0, 1) + 9g(1/3, 1/3)
24
τ̂
:=
4
X
wl g(x̂l ).
l=1
All integrals will be transferred to the reference triangle. Here we have
denoted the quadrature points on the reference element by {x̂l }, l = 1, 2, 3, 4.
Problem 1. Given a list of (hatx, haty) pairs (the quadrature nodes on τ̂ )
write a routine which computes the reference finite elements shape functions
and their gradients at each pair in the list, in the case of linears. Your
routine should have MATLAB interface:
function [hat_phi,hat_phix,hat_phiy]=FEEVAL(hatx,haty,nq)
Here x̂i = (x̂(i), ŷ(i)), i = 1, . . . , nq are pairs and hat phi, hat phix, hat phiy
are nq × 3 arrays returning the values for each basis function and point in
the list (of the basis function, its x derivative and its y derivative). The list
of (hatx, haty) pairs come from the quadrature.
Problem 2. Write a routine which given v1 , v2 , v3 computes the transformation matrix B appearing in the affine mapping which takes (0,0) to v1 ,
(1,0) to v2 and (0,1) to v3
1
2
Problem 3. Set up a routine which reads triangulation data generated by
“TRIANGLE” (this was discussed in recitation). Triangle gives the list of
global nodes (an ordered list of x-y pairs) and a list of elements (triangles).
Each element is identified by three global node indices (the global vertex
numbers of the triangle). This information provides the local to global node
map. You will have to create appropriate data structure to manage this
information (this was discussed in recitation). I have included a poly file for
running triangle on the L-shaped domain. The command for building the
finest mesh is
triangle − a.00001 − e − q30 ell.poly.
This results in a mesh with 238074 vertexes and 474127 triangles. The
“a” parameter above controls the triangle area, thus, for a mesh of twice the
mesh size, one would run a.00004. Generate six meshes starting from a.00001
using increases of 4 until the coarsest grid a.01024 (264 vertexes, 726 edges).
These are the meshes which you are to use to report computational results.
You can look at the coarser meshes using showme. It is also instructive to
run with a.2 and carefully examine the resulting output files ell.1.node and
ell.1.ele as the general version of these files will have to be read by your code.
Problem 4. After reading the mesh files, you need to assemble the global
stiffness matrix and global right hand side. The assembly code should have
the same structure as that used for the one dimensional problem and needs
to create a sparse matrix. This is easy in MATLAB as it supports sparse matrices. An example on initializing the matrix and subsequently incrementing
its entries is given in Wenyu’s 610 page. Note that you should initialize the
matrix with space for 9N non-zeros with N being the number elements. The
local stiffness matrices are to be approximated by changing variable to the
reference domain and applying the quadrature (discussed in class on Oct.
15).
Problem 5. Write a routine which generates local right hand side contributions and use these to assemble the right hand side vector. As above,
this involves only one loop through the elements and a change a variable so
that the computation can be approximated using the quadrature rule on the
reference element.
Problem 6. Approximate the solution of the problem
(6.1)
−∆u + 2u = (5π 2 + 2) cos(πx) cos(2πy)
∂u
= 0 on ∂Ω.
∂n
on Ω,
3
The weak formulation results in a bilinear form as in the Oct. 15 notes with
a(x) = 1 and q(x) = 2. Here Ω is still the L-shaped domain described earlier.
The solution of the above problem is u(x, y) = cos(πx) cos(2πy). Run your
program on mesh files (generated by triangle) using the calls
triangle
triangle
triangle
triangle
triangle
triangle
-a.01024
-a.00256
-a.00064
-a.00016
-a.00004
-a.00001
-e
-e
-e
-e
-e
-e
-q30
-q30
-q30
-q30
-q30
-q30
ell.poly.
ell.poly.
ell.poly.
ell.poly.
ell.poly.
ell.poly.
For each of the above meshes, generate the coefficient vector for the interpolant of the solution
X
X
vh =
u(xi )φi ≡
c i φi
i
i
where xi is the node associated with the finite element basis function φi (vh
is the finite element interpolant of theP
solution u). Compute the coefficients
of the finite element solution uh =
i di φi by solving the matrix vector
problem involving the stiffness matrix and the right hand side (computed
above). Report (for each mesh)
kuh − vh k = ((M (c − d)) · (c − d))1/2 .
Here k · k denotes the L2 (Ω) norm and M is the mass matrix, i.e.,
Z
φj φi dx.
Mij =
Ω
The mass matrix (in sparse form) can be assembled using your stiffness
matrix assembly code and setting a(x) = 0 and q(x) = 1. For one of the
meshes, plot the solution and approximate solution (separate graphs) and
turn copies of the figures.
Here are some hints on debugging your code. You should check your matrix
assembly routines on a simple mesh. I have included the node and ele files
for a simple triangulation of the unit square. For the first element of this
mesh, the local stiffness matrix for the Laplacian is:


1 −1 0
1
−1 2 −1 .
2
0 −1 1
4
Here the second unknown corresponds to the dof at the right angle. The
other local stiffness matrices are similar. The local mass matrix is given by


2 1 1
1 
1 2 1 .
96 1 1 2
(You should compute the above matrices by hand as an exercise). After
building your 2-d elements, verify that your local stiffness matrix is creating
the correct results in the above cases.